[R-sig-ME] Does lmer use Empirical Bayes to estimate random effects

[Douglas Bates]

On Thu, Oct 29, 2009 at 11:21 AM, Christopher David Desjardins
<desja004 at umn.edu> wrote:
> I noticed that HLM uses Empirical Bayes to estimate the random effects and I
> was curious if this is what lmer() does to? I came across this
> https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q1/002107.html
> but it is unclear to me if it is correct to say that lmer() uses Empirical
> Bayes to estimate these effects.

The short answer is "It depends on what you mean by Empirical Bayes
but probably yes".

The longer answer is that the term "Empirical Bayes" by itself is not
sufficiently specific to indicate a way to evaluate an "estimate".  It
is a description of a type of technique, not a statistical model per
se.

It is also symptomatic of a kind of fuzzy thinking that I would prefer
to avoid.  Technically the random effects are unobserved random
variables, not parameters, so it is questionable what one would mean
by estimates of the random effects.  I prefer to describe the values
returned by ranef as the conditional modes of the random effects given
the observed data values and the estimated parameter values.  In the
case of a linear mixed-effects model with underlying Gaussian
distributions the conditional modes are also the conditional means,
which may be easier to explain.